Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 1

Irvy M.A.Gledhill1, Hamed Roohani

2, Karl Forsberg

3, Peter Eliasson

3,

Beric W. Skews2 and Jan Nordström

4

Theoretical treatment of fluid flow for accelerating bodies

1 Aeronautical Systems Competency Area, Defence, Peace, Safety and Security Unit, CSIR,

Pretoria, 0001, South Africa 2 School of Mechanical, Aeronautical and Industrial Engineering, University of the

Witwatersrand, Johannesburg, 2050, South Africa 3 FOI, Totalförsvarets forskningsinstitut, The Swedish Defence Research Agency, 164 90,

Stockholm, Sweden 4

Division of Computational Mathematics, Department of Mathematics, Linköping University,

SE-581 83, Linköping, Sweden

Corresponding author:

Prof. I.M.A. Gledhill

DPSS

CSIR

PO Box 395

Pretoria 0001

South Africa

Email (preferred): igledhil@csir.co.za

Telephone: +27 12 841 2769

Mobile: +27 82 4139134

Fax: +27 12 349 1156

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 2

Abstract Most computational fluid dynamics simulations are, at present, performed in a body-fixed

frame, for aeronautical purposes. With the advent of sharp manoeuvre, which may lead to transient

effects originating in the acceleration of the centre of mass, there is a need to have a consistent

formulation of the Navier-Stokes equations in an arbitrarily moving frame. These expressions should

be in a form that allows terms to be transformed between non-inertial and inertial frames, and includes

gravity, viscous terms, and linear and angular acceleration. Since no effects of body acceleration appear

in the inertial frame Navier-Stokes equations themselves, but only in their boundary conditions, it is

useful to investigate acceleration source terms in the non-inertial frame. In this paper, a derivation of

the energy equation is provided in addition to the continuity and momentum equations previously

published. Relevant dimensionless constants are derived which can be used to obtain an indication of

the relative significance of acceleration effects. The necessity for using Computational Fluid Dynamics

to capture non-linear effects remains, and various implementation schemes for accelerating bodies are

discussed. This theoretical treatment is intended to provide a foundation for interpretation of

aerodynamic effects observed in manoeuvre, particularly for accelerating missiles.

Keywords fluid physics · Navier-Stokes equations · arbitrary acceleration · manoeuvre ·

Computational Fluid Dynamics · non-inertial frame

1 Introduction

Current aircraft and missile systems place increased reliance on high agility. Fifth generation missiles

turn at a maximum lateral acceleration of 60g to 100g, where g is the acceleration due to gravity.

Projectiles may be subjected to thrust of the order of 100g to 500g at launch, and thrust vectoring is

commonly used for air-to-air missiles. As a result, there is an increasing need for a consistent set of

Navier-Stokes equations in arbitrarily moving frames as a basis for further development in the field of

accelerating aerodynamics, specifically to assist in understanding the dynamics of manoeuvring air

vehicles in conjunction with CFD. The set of equations should cover subsonic, transonic, and

supersonic regimes, and provide insight on the physics of fluids in accelerating frames. It should define

scalar, vector and tensor properties in inertial and non-inertial frames, and show clearly which

coordinate systems are being used in either. The equations should include both viscous and

gravitational terms as well as linear or turning acceleration. In this paper the energy equation for

arbitrary acceleration in general non-inertial frames of reference is presented, based on the work of

Löfgren (unpublished reports FFAP-A-1040 and FFAP-B-066, The Aeronautical Research Institute of

Sweden, 1995 and 1998 respectively) and Forsberg (unpublished report, The Aeronautical Research

Institute of Sweden, 1998) and following the continuity and momentum conservation equations

previously formulated by Gledhill et al. [1]. From the accelerating frame formulation, a set of

dimensionless constants is derived, which, under specified circumstances, can be used to determine

whether acceleration effects are likely to be significant in a specified regime of interest in terms of the

relative importance of linear terms.

To date, most aeronautical design, and guidance and control, has been based on steady-state

analysis, supplemented by dynamic derivatives, with the notable exception of aeroelastic

configurations. However, an increasing number of applications now require an understanding, not

simply of transient effects, but of the aerodynamics of significantly accelerating objects. The examples

of interest for this paper are rapidly manoeuvring missiles, missiles under significant unidirectional

acceleration, and projectiles experiencing spin-down or spin-up. In these cases, flow changes are

observed over a trajectory which is orders of magnitude longer than the vehicle itself.

Added mass effects are important for manoeuvre for vehicles suspended in a medium of similar

density to their own, such as airships and submarines. In some of these cases, the vehicle is moving

within its own wake, and the flow effects are important within a domain of the same order of size as the

vehicle itself. In the present work, the emphasis is on arbitrary manoeuvre over a path that is longer

than the typical length of the vehicle, although the theory can be applied when it is relevant to local

effects such as periodic motions, aeroelastic motions, rotors, oscillating wings, or effects which can be

modelled as locally unsteady without considering acceleration of the centre of mass of the vehicle.

This paper is intended to set out the relevant fundamental theory in a form useful for finite-

volume implementation. For that reason, illustrations have been given in terms of examples that have

already been published by our group and others. The restricted case of acceleration along a straight line

is described, and dimensionless parameters for example configurations are illustrated.

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 3

2 Earlier work

Turbine engine design, and geophysical flows, have been the primary fields in which rotating flow has

been of interest, and formulations of equations for constant angular velocity flow are to be found in

fundamental textbooks [2,3]. The theory of rotating fluids was explored by Hough [4], Proudman [5],

Taylor [6,7] and Boubnov and Golitsyn [8], and is summarised by Greenspan [9] and Owczarek [10],

using derivations based on flow volumes in contrast to the current general transforms.

Several early authors performed analytical studies of Stokes flow over bodies accelerating in

incompressible viscous flow at low Reynolds numbers, deriving the force on a sphere and an

expression for the boundary layer. This led to the formulation of the Basset-Bousinessq-Oseen equation

for the motion of a particle in the limit of low Mach and Reynolds numbers and its subsequent

extension to compressible flow [11]. The terms Bousinessq-Basset force, Basset force [12] or history

force refer to the fact that the development of a boundary layer over an accelerating particle in a fluid

depends on the temporal history of the fluid near the particle surface. Boundary layer start-up processes

with impulsive or finite acceleration were studied analytically by Blasius [13], Görtler [14] and

Schlichting [15], and similar studies have been performed for spheres and spinning spheres [16,17,18]

for constant acceleration or falling under gravity. Renewed interest in boundary layers with

accelerating or decelerating free-stream flow (as in a converging flow) has led to many studies, e.g.

[19,20,21], of flow within favourable pressure gradients, and of relaminarisation. This work is relevant

to understanding the behaviour of turbulent boundary layers at accelerating walls if acceleration effects

are small enough for Galilean invariance to be assumed.

The requirement for an understanding of noise from supersonic aircraft prompted a geometrical

analysis of Mach and shock waves, supported by towing-tank experiments, by Lilley et al. [22].

A potentially useful direction in terms of simulation is that of deriving invariant schemes for

moving frames based on numerical considerations, and is largely based on the work of Olver [23]

involving Lie group transformations of partial differential equations. Poludnenko and Khokhlov [24]

required a simulation system for astrophysics and cosmology which would be suitable for cases where

the typical flow speeds are very much larger than the sound speed or the local flow velocities, in, for

example, stellar cores or galactic formation. The inviscid equations are derived using a transformation

into a non-inertial rotating frame with scaling factors in time, space, and density for this purpose. The

computational frame transform is chosen to be reasonably close to the expected stellar or galactic

evolution, including oscillatory motions. In a similar way Chhay and Hamdouni [25] derived

symmetrisation parameters for the inviscid flow equation transformation groups. These methods are

particularly useful when expansion and contraction is very significant rather than centre of mass

acceleration.

Understanding of added mass effects has also depended on the history force concept [26] and

has been reviewed and measured in low Reynolds number experiments with gravitational acceleration

[27]. A drag calculation for a manoeuvring underwater vehicle was worked out from first principles by

Sonowski et al. [28] for use in a navigation, object detection and collision avoidance system, and

performed well in a simulated environment.

In terms of dimensionless numbers which may aid in defining flow regimes, work from the area

of tunnel startup by Freymuth et al. [29], two useful dimensionless parameters were introduced: a

dimensionless time, 𝑡∗ =𝑡

√𝑎/𝐿, where L is a typical length and the normalising time is the typical time

for air to cross length 𝐿 from rest at constant acceleration 𝑎; and a Reynolds number in acceleration,

𝑅 =𝐿√𝐿𝑎

𝜈, based on the final velocity after acceleration from rest, and the kinematic viscosity 𝜈.

Given the need for prediction of loads on manoeuvring missiles, Löfgren (ibid.) developed a

mathematical toolbox for arbitrarily moving frames of reference. This study was initiated with the

derivation of the properties of general linear transforms between arbitrarily moving frames, with

relative motion defined by functions of continuous differentiability of class C2, in Rn+1

spaces with n

spatial dimensions. Transformations for scalars, vectors and tensors, and operators, were defined, from

which transforms of derivatives were deduced. General conservation equations were written for

moving volumes, and then reduced to three spatial dimensions. The equations for particle motion, and

for fluid conservation, could then be written. The weak variants of the differential equations of fluid

flow were examined assuming local differentiability in the solutions, and jump conditions at

discontinuities were derived. The integral forms of the conservation equations were derived for moving

control volumes. Any two arbitrarily moving frames were treated to this point in a symmetrical way,

and from these results conservation relations for mass, momentum and energy were derived

specifically for inertial and non-inertial frames. Forsberg (ibid.) then formulated the mass, momentum

and energy conservation equations in preparation for CFD implementation. Forsberg’s derivation of the

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 4

mass and momentum conservation relations appeared in 2003 [30] together with CFD implementation

and applications. The concept of generalised rothalpy was introduced by Forsberg, and is provided

here, together with Forsberg’s derivation of the energy equation, which is now extended to include

external forces in order to demonstrate the relationship between gravitational effects and the effects of

an arbitrarily accelerating frame origin.

Gledhill et al. [1] published the continuity and momentum equations and demonstrated an

implementation in the code EURANUS [31], by performing the computations in the inertial frame. This

approach has the considerable advantages of (1) a very simple development path from an existing

moving-mesh model with spatial conservation [31], in which boundary conditions are the main

modifications required, and (2) the fact that the source terms appearing in the relative frame model

(below) do not need to be discretised, implemented, verified or validated. It is known that source terms

such as those required for acceleration can make the numerical system stiff.

Limache [32,33] approached the problem using the methodology of Owczarek [10] specifically

for aerodynamically steady motions, motivated by the potential for using these as a set of characteristic

motions for the description of general manoeuvre. The Navier-Stokes equations were generalised for

aerodynamically steady flow, and non-dimensionalised so that coefficients could be written in terms of

Reynolds, Prandtl and Mach numbers. Non-dimensional coefficients were not, however, explored for

characterisation of the significance of acceleration. A method of derivation of the Coriolis effect in the

momentum equation has been described by Kageyama and Hyodo [34], using Galilean transformation

between successive inertial frames. The method ultimately leads to a momentum equation equivalent to

those derived by Batchelor and Greenspan [3,9]. Gardi [35] presented similar momentum and energy

terms as source terms for a non-inertial moving reference frame and discussed discretisation errors in

comparison with an Arbitrary-Lagrangian-Eulerian approach. Combrinck and Dala [36] extended the

procedure of Kageyama and Hyodo [34] to transforming the Navier-Stokes equations under pure

rotation, for constant and varying angular velocity, using successive transforms between an inertial

frame, an orientation preserving frame, and a rotated frame. The two moving frames have the same

angular velocity. It was demonstrated that internal energy conservation is invariant. The results are

shown for compressible and incompressible flow.

The work in this paper extends the theoretical endeavour described above to expressions for

generalised rothalpy, the derivation of dimensionless constants specifically for the purpose of

characterising regimes in accelerating flight, and the illustration of these parameters for practical cases.

CFD calculations on accelerating aerofoils or missiles have been carried out by very few

authors. Inoue et al. [37] used an early simulation in the inertial frame to simulate a constant speed

diamond aerofoil describing a parabolic path, for the purpose of investigating shock focussing within

the arc. Forsberg et al. [30] and Gledhill et al. [1] provided the basis of the theory for the continuity

and momentum equations as given here, validated an implementation in the inertial frame within the

finite-volume code EURANUS using the oscillating aerofoil data of Landon [38], and then used a

model of the NACA0012 aerofoil on constant pitch trajectories [39] to illustrate the necessity for

inclusion of acceleration effects in the calculation of pitch damping. It was then demonstrated that drag

on an axisymmetric flare missile dropped by 20% in comparison with steady values if the flare

geometry was accelerated through the transonic range at 4 500 ms-1

.

Roohani and Skews have conducted detailed studies of the unsteady aerodynamics of

accelerating bodies, such as aerofoils [40-44] and axisymmetric objects [45] undergoing unidirectional

acceleration through the subsonic and transonic ranges. This was modelled numerically using source

terms added to the momentum and energy equations in the code FLUENT [46].Shock wave boundary

layer interaction was found to have significant effect on the shock wave position and strength in the

transonic range. Comparison of lift, pitching moment, and drag, as well as shock and flow field

behaviour in acceleration, deceleration and steady state at specific Mach numbers during flight

revealed significant differences between the steady and unsteady scenarios. Acceleration dependent

behaviour for bow shocks, tail shocks and trailing compression waves were observed. In conclusion, all

differences between steady and unsteady transonic shock wave behaviour and lift were found to be

predominantly a function of flow history, while differences between steady and unsteady subsonic drag

were mainly due to fluid inertia. It was shown that the angle at which separation occurs at a specific

subsonic Mach number varies between the steady, accelerating and decelerating scenarios [47].

At this point it is important to explain the concept of flow history as used in the present context

[43]. When an object is accelerated in a compressible fluid, the unsteady flow field generated is not

allowed sufficient time to reach equilibrium at any specific Mach number. It is therefore different from

the steady state flow field at that same Mach number and is in fact influenced by its own earlier flow

field characteristics. During acceleration, subsonic lift values are found to be lower than the

corresponding steady state values at the same Mach number because the unsteady flow field has

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 5

insufficient time to reach equilibrium and will reflect some of the earlier characteristics when the Mach

number was lower.

In terms of significant angular velocity, the findings of Naidoo and Skews [48,49] for the flow

field about a rapidly accelerating wedge causing shock reflection off a wall showed hysteresis effects in

transition between regular and Mach reflection. These effects take place within a local domain, and

accurate observations from wind tunnel experiments were made and compared with moving-mesh

CFD.

Experimental data for validation purposes is rare for long trajectories, but ballistic ranges can

provide some data. The application of CFD to prediction of the characteristics of army projectiles was

initiated at Aberdeen Proving Ground for calculations of pitch damping force and moment coefficients

[50,51]. As a method of computing flow fields in deceleration for comparison with ballistic range data,

Saito et al. [52] used an inertial frame CFD system for models of shock stand-off distance.

In practical aeronautical terms, since it is of considerable importance to predict dynamic

derivatives for missiles and projectiles, and to some extent these needs have driven the field of

manoeuvre modelling forward [53,54,55]. Sturek et al. [56], Wienacht [51], Murman et al. [57], and

McIlwain et al. [58] have employed prescribed-trajectory methods for missiles in local domains. The

coupling of time-accurate CFD with rigid body dynamics allows the prediction of trajectories, but

computational resources are not yet sufficient to make this viable for more than fractions of a second

[59]. The use of flight dynamic databases through which a configuration is “flown” has been developed

in aircraft manoeuvre modelling, and for complex non-linear manoeuvres requires the addition of

dynamic derivatives [60] and, as a next step, Reduced-Order Methods (e.g. [61]). These have proved

reasonable for linear coefficients, but less accurate for non-linear rolling and yawing moments.

Given the above, there is a need for a theoretical basis and a complete set of conservation

equations in both inertial and non-inertial frames on which further work can be founded, particularly

for manoeuvre over extended trajectories. Transformation of the continuity and momentum equations

[1] is extended here to the energy equation. Part of the motivation for developing this theory is to

develop guidance about what constitutes “significant” acceleration, and how the regimes of flow about

accelerating objects can be distinguished and quantified. As a step in this direction, some

dimensionless constants are derived. Where data is available, these constants have been calculated, and

some remarks about their applicability are made. A further reason for presenting the theory is to assist

in supporting decisions about the relative merits of numerical models in inertial and non-inertial

frames.

The approach presented here is given in terms of familiar formulations of the conservation

equations that are frequently used in aerospace applications. The distinctions between expression of

these equations in inertial and non-inertial frames is clearly made. The theory given here has the

advantages of allowing transformation between body-fixed coordinates attached to manoeuvring

vehicles, and inertial frames, in which flow fields and flow phenomena are often unfamiliar and harder

to interpret. Many of the source terms arising in the momentum equation are familiar, and are given

here in a general and inclusive way for all the effects of arbitrary trajectories. The energy equation is

presented in a form from which cross-terms can be identified, and, in future, quantified.

3 Theory

A mathematical toolbox for the treatment of non-stationary problems with moving boundaries, for

application in fluid dynamics and aeronautics, was formulated by Löfgren (ibid.). Löfgren developed

an algebraic basis for a finite-dimensional vector space, in which linear transformations relevant to

fluid balance equations could be defined. Space and time were treated on an equal footing as far as

possible. The properties of functions and derivatives under transformation were developed, and the

conservation equations were constructed in arbitrarily moving control volumes. The conservation laws

were expressed in terms of displacement vectors 𝒙 and time 𝑡, for density 𝜌 = 𝜌(𝒙, 𝑡), velocity

𝒗 = 𝒗(𝒙, 𝑡), pressure 𝑝 = 𝑝(𝒙, 𝑡), temperature 𝑇 = 𝑇(𝒙, 𝑡), and an external force per unit mass

𝒈 = 𝒈(𝒙, 𝑡), in two arbitrarily moving frames of reference Σ and Σ′. A generalised conservation

equation in its strong integral form was treated for an arbitrarily moving volume, and the weak

differential form was developed. One of the frames was then constrained to be an inertial frame, in

which Newton’s Laws hold, and are expressed in the form of conservation equations. The transforms to

the arbitrarily moving frame Σ' were derived for two dynamic systems: for a finite mass particle, and

for a homogeneous fluid. Löfgren was able to prove the hypothesis that “for a fluid in space, the same

balance equations hold in Σ' as in Σ, provided that in Σ′ we add the inertial terms per mass unit which

would be added in the case of a particle”.

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 6

This work established a sound mathematical foundation for the expression of the method in a

form suitable for making choices between possible numerical implementations in an inertial frame or in

non-inertial frames by Forsberg (ibid.), which forms the basis for the analysis given here.

In Forsberg’s formulation, vectors are expressed in an inertial frame and in an arbitrarily

moving frame. Since the latter may also be an inertial frame if it moves with constant velocity relative

to the inertial frame, we sometimes refer to it as the relative frame, and to the inertial frame as the

absolute frame.

Any convenient inertial frame may be chosen as the absolute frame Σ. For aeronautical

purposes, it has been useful to illustrate the inertial frame as being fixed to “the ground”. This frame is

not in fact inertial, since it is rotating and moving with the earth, and is in a gravitational field. The

inertial frame in our analysis can be defined as being stationary with respect to an infinite domain of

still air, and gravity is accommodated as a body force in the equations of motion without loss of

generality.

The relative frame Σ′ may be chosen as, for example, fixed to a manoeuvring missile in such a

way that the orientation of the body axes follows that of the object. CFD has usually been carried out

on a mesh in this frame, and for this reason, flow fields about objects have traditionally been

interpreted in this frame. In the relative frame Σ′, a stagnation point with zero velocity relative to the

missile appears at the leading edge or nose, and non-zero values of the velocity are specified at the far-

field boundaries of the finite domain mesh. In contrast, if the flow is simulated by a mesh attached to

the object moving through the absolute frame and computations are performed in the absolute frame,

the far-field boundaries of the moving mesh encounter stagnant air, and therefore the far-field boundary

condition on each velocity component is zero. The wall at the leading edge of the flying object is

travelling with a non-zero time-dependent velocity. The velocity boundary conditions of the missile

wall are determined by the dynamics of the flight, and the velocity of air in close proximity to the

leading edge will be non-zero. Near solid non-slip walls in viscous flow, the boundary layer will

exhibit velocities in the viscous sublayer close to those determined by the moving wall dynamics.

Once practical models are introduced, it is useful to provide a general transform between frames

for ease of interpretation of familiar flow features, and for identification of new ones. This

transformation gives rise to the phrases “as seen from the absolute frame” or “as seen from the relative

frame” below. These refer to vector components with respect to a set of orthonormal basis vectors {𝑒1, 𝑒2, 𝑒3} or {𝑒1

′ , 𝑒2′ , 𝑒3

′ } .

3.1 Notation and transforms

Consider an inertial frame Σ (Figure 1(a)) and a non-inertial frame Σ' (Figure 1(b)). We establish the

following notation for vectors and tensors in these frames:

𝒂 is a vector in Σ described by coordinates in Σ,

�̆� is a vector in Σ described by coordinates in Σ',

𝒂 is a vector in Σ' described by coordinates in Σ', and

�̂� is a vector in Σ' described by coordinates in Σ.

In summary, the underscore notation describes a vector in the relative frame, while the breve

( ̆ ) and circumflex ( ̂ ) are used to indicate transformed vectors. As an example, the components of

gravity vector 𝒈 expressed with respect to a set of basis vectors in Σ are constant in time, but if Σ'

rotates, the equivalent vector �̆� expressed in terms of a basis set in Σ' will have time-dependent

components. The mapping between Σ and Σ′ back is defined by a unitary transform 𝑈 defined in the

following way:

�̂� = 𝑼 ∙ 𝒂 (1)

Then

�̆� = 𝑼−1 ∙ 𝒂 = 𝑼𝑡 ∙ 𝒂 (2)

�̆̂� = 𝒂 (3)

and

det(𝑼) = 1. (4)

Here 𝑼 may be interpreted as a rotation such as

𝑼 = [

𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃 0𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 0

0 0 1] (5)

where 𝜃 is an axis angle, in this case with respect to 𝑒3. Tensors are transformed by

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 7

�̆� = 𝑼𝒕 ∙ 𝝉 ∙ 𝑼, 𝝉 = 𝑼 ∙ �̆� ∙ 𝑼𝒕 (6)

We now need to relate position vectors in the two frames. A position vector in the inertial frame is

denoted by 𝒙, and we define the transform relating 𝒙 in the relative frame to 𝒙 by

𝒙 = 𝒓 + 𝑼 ∙ 𝒙 and 𝒙 = 𝒓 + 𝑼𝒕 ∙ 𝒙. (7)

Here 𝒓(𝑡) is the position vector of 𝑂′, the origin of Σ′, relative to 𝑂, the origin of Σ. As an

example, 𝒙 may denote a position vector of a point P relative to body axes with origin 𝑂′ fixed in the

body, and 𝒙 is the position vector of P with respect to 𝑂. We note for 𝒙 = 0 that 𝒓 = −𝑼𝒕 ∙ 𝒓 and

𝒓 = −𝑼 ∙ 𝒓. For acceleration cases we require that 𝒓(𝑡) is 𝐶2 regular in time, i.e. that the second

derivative with respect to time is continuous and smooth. Note that 𝒓 may be used to describe

revolution of 𝑂′about 𝑂, while 𝑼 and 𝝎 (below) describe spin or rotation about 𝑂′. The two angular

displacements and velocities need not be the same.

(a) (b)

Figure 1 (a) Position vectors seen from Σ at successive times t1 and t2, (b) Position vectors seen

from Σ' at successive times t1 and t2

For time derivatives it is convenient to define a vector 𝝎 by

𝝎 × (𝑼 ∙ 𝒂) =

𝜕𝑼

𝜕𝑡∙ 𝒂 (8)

Differentiating with respect to time (and using dot notation for time derivatives) we obtain

absolute and relative velocities respectively:

𝒗 =

𝑑𝒙

𝑑𝑡=

𝑑𝒓

𝑑𝑡+

𝜕

𝜕𝑡𝑼 ∙ 𝒙

= �̇� + 𝑼 ∙ (�̆� × 𝒙 + 𝒗)

= �̇� + �̂� + 𝝎 × 𝒙

(9)

(10)

Similarly

𝒗 = −�̇� + �̆� − �̆� × 𝒙. (11)

An inter-frame or system point velocity field 𝒖 between the two frames is defined by

𝒖 = �̆� − 𝒗 (12)

which leads to an interframe acceleration term

�̇� = �̇̆̇� + �̇̆� × 𝒙 (13)

where × denotes a vector cross product. We can also show for a general vector 𝑎 that

𝒂 ⋅ 𝛁𝒖 = �̆� × 𝒂. (14)

3.2 Conservation equations

We are interested in deriving conservation laws expressed in Σ′ for several purposes. The first is that a

number of effects in a non-inertial frame are familiar, such as the Coriolis effect, and should be shown

in the context of flow around accelerating bodies; the second is that terms for linear acceleration and

angular acceleration in rotation and revolution should be identifiable; and the third is to illustrate the

use of dimensionless parameters in this field. In a previous paper, the derivation of the conservation

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 8

equations for mass and momentum was presented by Gledhill et al. [1]. In the following paragraphs,

the mass and momentum equations are summarised, and the energy equation is derived.

The aim is to start with the conservation equations in Σ, expressed in terms of 𝒗, and write them

in terms of 𝒖 and 𝒗, the velocity in Σ′ with respect to coordinates in Σ′. In the latter form, without any

acceleration, they correspond to the solution commonly used in external CFD. Without acceleration,

both frames are inertial, and the equations take the same form for 𝒗 and Σ as they do for 𝒗 and Σ′.The

intrinsic functions 𝜌, 𝑝, 𝑒 and 𝑇 are Galilean invariant and undergo a change of variables. Since 𝒓 is

easy to transform, �̆� may be used. Some parameters, such as the stress tensor, 𝝉, are not the focus of the

current development, but the fact that they are written in terms of coordinates in Σ′, e.g. �̆� , allows the

form in which they have been included to be clearly identified for further expansion.

3.3 General conservation laws

Mappings 𝜙: ℝ1+3 → ℝ1+3 defined by 𝜑(𝑡, 𝒙) = (𝑡, 𝒓 + 𝑼 ⋅ 𝒙), and the inverse 𝜑∗(𝑡, 𝒙) = (𝑡, −�̆� +

𝑼𝒕 ⋅ 𝒙), are performed. Then conservation equations may be written in the absolute frame Σ in strong

form as follows, where the symbol 𝒂 may represent a scalar (an element of ℝ1), or a vector (an element

of ℝ3), for a moving volume 𝑉:

𝜕

𝜕𝑡∫ 𝜌𝒂𝑑𝑉

𝑉

+ ∮ (𝜌𝒂�̂� + 𝑭𝑎) ∙ 𝑑�̂� = ∫ 𝑄𝑎𝑑𝑉𝑉𝜕𝑉

(15)

where 𝑄𝑎 is the volume source term of 𝒂, 𝑭𝑎 is a flux term, and 𝑑𝑺 are surface normals, which can be

defined in a natural way in the reference frame as 𝑑�̂� . The volume element 𝑑𝑉 is conserved across

frames by the definition of 𝑼. To express this equation in the relative frame Σ′, the mapping 𝜑∗ is

applied:

𝜕

𝜕𝑡∫ 𝜙∗(𝜌𝒂)𝑑𝑉

𝑉+ ∮ (𝜙∗(𝜌𝒂�̂�) + 𝜙∗𝑭𝑎) ∙ 𝑑𝑺 = ∫ 𝜙∗𝑄𝑎𝑑𝑉

𝑉𝜕𝑉. (16)

The functions 𝜌, 𝑝, 𝑇 and 𝑄𝑎 can be treated as scalar functions with the change of variable specified by

𝜑∗ . Then

𝜕

𝜕𝑡∫ 𝜙∗(𝜌𝒂)𝑑𝑉

𝑉+ ∮ [𝜙∗(𝜌𝒂)(𝜙∗𝒗 − 𝒖) + 𝜙∗𝑭𝑎] ∙ 𝑑𝑺 = ∫ 𝜙∗𝑄𝑎𝑑𝑉

𝑉𝜕𝑉. (17)

Now it is possible to regard the scalar functions as defined in Σ′ and omit 𝜙∗ for convenience to write,

in Σ′,

𝜕

𝜕𝑡∫ 𝜌𝒂𝑑𝑉

𝑉

+ ∮ (𝜌𝒂𝒗 + �̆�𝑎) ∙ 𝒅𝑺 = ∫ 𝑄𝑎𝑑𝑉𝑉𝜕𝑉

(18)

Using Green’s theorem, the differential form in the relative frame can be written

𝜕

𝜕𝑡(𝜌𝒂) + 𝛁 ∙ (𝜌𝒂𝒗 + �̆�𝑎) = 𝑄𝑎 (19)

Alternatively, equation (19) can be proved by writing the forms of derivatives under 𝜑∗.

3.4 Conservation of mass

The mass continuity equation is obtained with scalar 𝑎 = 1, 𝑭𝑎 = 0 and 𝑄𝑎 = 0 in the absence of

sources or sinks. In the absolute frame:

𝜕

𝜕𝑡∫ 𝜌𝑑𝑉

𝑉

+ ∮ 𝜌�̂� ∙ 𝑑�̂� = 0𝜕𝑉

(20)

or equivalently in the relative frame

𝜕

𝜕𝑡∫ 𝜌𝑑𝑉

𝑉

+ ∮ 𝜌𝒗 ∙ 𝒅𝑺 = 0𝜕𝑉

(21)

or

𝜕𝜌

𝜕𝑡+ 𝛁 ∙ (𝜌𝒗) = 0. (22)

3.5 Conservation of momentum

For the momentum conservation equation, 𝒂 = 𝒗 in the inertial frame, 𝑭𝑎 = 𝑝𝑰 − 𝝉, where 𝑰 is the

identity matrix of rank three, and 𝑄𝑎 = 𝜌𝒈 in the absence of other external forces [2]:

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 9

𝜕

𝜕𝑡∫ 𝜌𝒗𝑑𝑉

𝑉

+ ∮ (𝜌𝒗⨂𝒗 + 𝑝𝑰 − 𝝉) ∙ 𝒅�̂� = ∫ 𝜌𝒈𝑑𝑉𝑉𝜕𝑉

(23)

An outer product of tensors is required here. For a tensor Θ, with components 𝜃𝑖1…𝑛, outer

products are defined by [Θ⨂Ψ]𝑖1…𝑖𝑛,𝑗1…𝑗𝑚= 𝜃𝑖1…𝑖𝑛

𝜓 𝑗1…𝑗𝑚 and 𝑈 ∙ (Θ⨂Ψ) = (𝑈 ∙ Θ)⨂(𝑈 ∙ Ψ). To

transform to the relative frame, it is therefore possible to write

𝜕

𝜕𝑡∫ 𝜌�̆�𝑑𝑉

𝑉+ ∮ (𝜌�̆�⨂𝒗 + 𝑝𝑰 − �̆�) ∙ 𝒅𝑺 = − ∫ (𝜌�̆� × �̆� − 𝜌�̆�)𝑑𝑉

𝑉𝜕𝑉. (24)

To express this in differential form we note that, owing to the definitions of scalar products and

divergence operators [𝚯 ∙ 𝚿]𝑖1…𝑖𝑛−1,𝑗2…𝑗𝑚= ∑ Θ𝑖1…𝑖𝑛−1𝑘Ψ𝑘,𝑗2…𝑗𝑚𝑘 and [𝛁 ∙ 𝚯]𝑖2…𝑖𝑛

= ∑𝜕Θ𝑘𝑖2…𝑖𝑛

𝜕𝑥𝑘𝑘 , it is

correct to write

∮ (�̆�⨂𝒗) ∙ 𝑑𝑺 = ∫ ∇ ∙ (𝒗⨂�̆�)𝑑𝑉𝑉𝜕𝑉

and therefore

𝜕

𝜕𝑡(𝜌�̆�) + 𝛁 ∙ (𝜌𝒗⨂�̆� + 𝑝𝑰 − �̆�) = −𝜌�̆� × �̆� + 𝜌�̆�. (25)

Note that in a rotating frame, the gravity vector �̌� will rotate. Using the method of Forsberg (ibid.) and

Gledhill et al. [1] to write the expression in terms velocities �̆� expressed in Σ′, (22) becomes

𝜕

𝜕𝑡∫ 𝜌𝒗𝑑𝑉

𝑉

+ ∮ (𝜌𝒗⨂𝒗 + 𝑝𝑰 − �̆�) ∙ 𝑑𝑺 = − ∫ 𝜌(�̇� + 2�̆� × �̆� + �̆� × �̆� + �̆�)𝑑𝑉𝑉𝜕𝑉

= − ∫ 𝜌[�̆� × ( �̆� × 𝒙) + 2�̆� × 𝒗 +𝜕�̆�

𝜕𝑡× 𝒙 +

𝜕�̆̇�

𝜕𝑡− �̆�]𝑑𝑉

𝑉.

(26)

In the relative frame in differential form this is

𝜕

𝜕𝑡(𝜌𝒗) + 𝛁 ∙ [𝜌𝒗⨂𝒗 + 𝑝𝑰 − �̆�]

= −𝜌�̆� × ( �̆� × 𝒙) − 2𝜌�̆� × 𝒗 − 𝜌�̇̆� × 𝒙 − 𝜌�̇̆̇� + 𝜌�̆�.

(27)

The source terms on the right hand side are easily identified as fictitious effects [9,2,3]: respectively,

they are the centrifugal effect, the Coriolis effect, the Euler term for spin-up or spin-down [1], and

translational acceleration. The final term on the right hand side is the effect of a body force on the fluid,

in this case gravity.

3.6 Conservation of energy

To apply the general conservation relation for energy, we define 𝒂 = 𝐸, the total energy per unit mass,

and 𝑭𝑎 = 𝑝𝒗 − 𝝉 ∙ 𝒗 − 𝜅𝛁𝑇. We note from the work of Löfgren that 𝜌, 𝑝 and 𝑇 are transformed by 𝜙∗

as above, and �̆� and 𝝉 are an inertial frame vector and tensor respectively, expressed in coordinates of

the relative frame. The total energy per unit mass is related to the internal energy per unit mass by

𝐸 = 𝑒 +‖𝑣‖2

2. (28)

A heat source is included as 𝑞𝐻 and gravity is again applied as an external force acting on the fluid in

the control volume dV [2],

𝑄𝑎 = 𝑞𝐻 + 𝜌𝒗 ∙ 𝒈. (29)

In the absolute frame

𝜕

𝜕𝑡∫ 𝜌𝐸𝑑𝑉

𝑉

+ ∮ (𝜌𝐸�̂� + 𝑝𝒗 − 𝝉 ∙ 𝒗 − 𝜅𝛁𝑇) ∙ 𝒅�̂� = ∫ (𝑞𝐻 + 𝜌𝒈 ∙ 𝒗)𝑑𝑉𝑉𝜕𝑉

(30)

and in the relative frame

𝜕

𝜕𝑡∫ 𝜌𝐸𝑑𝑉

𝑉+ ∮ (𝜌𝐸𝒗 + 𝑝�̆� − �̆� ∙ �̆� − 𝜅𝛁𝑇) ∙ 𝒅𝑺 = ∫ (𝑞𝐻 + 𝜌�̆� ∙ �̆�)𝑑𝑉

𝑉𝜕𝑉. (31)

Expressed in terms of velocities 𝒗 in the relative frame on the left hand side, and using equation (12),

𝜕

𝜕𝑡∫ 𝜌𝐸𝑑𝑉

𝑉

+ ∮ (𝜌𝐸𝒗 + 𝑝𝒗 − �̆� ∙ 𝒗 + 𝑝𝒖 − �̆� ∙ 𝒖 − 𝜅𝛁𝑇) ∙ 𝑑𝑺𝜕𝑉

= ∫ (𝑞𝐻 + 𝜌�̆� ∙ �̆�)𝑑𝑉𝑉

.

(32)

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 10

The formulation of a generalised rothalpy conservation expression is now pursued. To rewrite

in terms of 𝒗 , equation (12) is used and it is noted that since 𝒖 = �̆̇� + �̆� × 𝒙,

�̆� ∙ �̆� = 𝒗 ∙ �̆� + �̆̇� ∙ �̆� + (�̆� × 𝒙) ∙ �̆� = 𝒗 ∙ �̆� + �̆̇� ∙ �̆� − 𝒙 ∙ (�̆� × �̆�). (33)

To relate internal energy and velocity �̆� through rothalpy, the scalar product of the momentum

conservation relation, equation (25), with the inter-frame velocity is formed:

𝝏

𝝏𝒕(𝜌�̆�) ∙ 𝒖 + {𝛁 ∙ [𝜌𝒗⨂�̆� + 𝑝𝑰 − �̆�]} ∙ 𝒖 = −𝜌(�̆� × �̆�) ∙ 𝒖 + 𝜌�̆� ∙ 𝒖 (34)

Since

𝜕

𝜕𝑡(𝜌�̆� ∙ 𝒖) = (

𝜕

𝜕𝑡𝜌�̆�) ∙ 𝒖 + 𝜌�̆� ∙ �̇�, (35)

equation (34) becomes

𝜕

𝜕𝑡(𝜌�̆� ∙ 𝒖) + {𝛁 ∙ [𝜌𝒗⨂�̆� + 𝑝𝑰 − �̆�]} ∙ 𝒖 = −𝜌(�̆� × �̆�) ∙ 𝒖 + 𝜌�̆� ∙ �̇� + 𝜌�̆� ∙ 𝒖. (36)

Using the definition of the inter-frame velocity, it is not difficult to show that 𝛁𝒖 is an anti-symmetric

tensor, and that its inner product with any symmetric tensor is 0, the inner product being defined for

second rank tensors as Θ: Ψ = ∑ 𝜃𝑖𝑗𝜓 𝑖𝑗𝑖𝑗 . Thus in the 4th

term above

�̆�: 𝛁𝒖 = 0 ,

𝛁 ∙ (�̆� ∙ 𝒖) = (𝛁 ∙ �̆�) ∙ 𝒖 + �̆�: 𝛁𝒖 = (𝛁 ∙ �̆�) ∙ 𝒖 . (37)

In the second term above, an inner product must be formed, and we note that (𝒗⨂𝒗) ∶ 𝛁𝒖 = 0. From

the definitions of scalar, outer and inner products and the inter-frame velocity 𝒖, a number of vector

relations arise, in addition to which

𝛁 ∙ (𝑝𝒖) = (𝛁𝑝) ∙ 𝒖 = [𝛁 ∙ (𝑝𝑰)] ∙ 𝒖 (38)

and

𝛁 ∙ [𝒗(�̆� ∙ 𝒖)] = [𝛁 ∙ (𝒗⨂�̆�)] ∙ 𝒖 + (�̆� × 𝒗) ∙ 𝒖. (39)

Now (36) can be expressed in terms of velocities defined in the relative frame 𝒗 and 𝒖:

𝜕

𝜕𝑡(𝜌�̆� ∙ 𝒖) + {𝛁 ∙ [𝜌𝒗⨂�̆� + 𝑝𝑰 − �̆�]} ∙ 𝒖

= −𝜌(�̆� × �̆�) ∙ 𝒖 + 𝜌�̆� ∙ �̇� + 𝜌�̆� ∙ 𝒖𝜕

𝜕𝑡(𝜌�̆� ∙ 𝒖)

+[𝛁 ∙ (𝜌𝒗⨂�̆�)] ∙ 𝒖 + 𝛁 ∙ (𝑝𝒖) − 𝛁 ∙ (�̆� ∙ 𝒖) − 𝜌(�̆� × �̆�) ∙ 𝒖 + 𝜌�̆� ∙ �̇� + 𝜌�̆� ∙ 𝒖

(40)

Using (3)

𝜕

𝜕𝑡(𝜌�̆� ∙ 𝒖) + 𝛁 ∙ [𝜌𝒗(�̆� ∙ 𝒖) + 𝑝𝒖 − �̆� ∙ 𝒖] = 𝜌�̆� ∙ �̇� + 𝜌�̆� ∙ 𝒖 (41)

By subtraction of equation (41) from the differential form of equation (32),

𝝏

𝝏𝒕𝝆(𝑬 − �̆� ∙ 𝒖) + 𝛁 ∙ [𝝆𝒗(𝑬 − �̆� ∙ 𝒖) + 𝒑𝒗 − �̆� ∙ 𝒗 − 𝜿𝛁𝑻]

= 𝑞𝐻 + 𝜌𝒗 ∙ �̆� − 𝜌�̆� ∙ �̇� (42)

It is now convenient to introduce a generalized rothalpy E* defined by

𝐸∗ = 𝐸 − �̆� ∙ 𝒖 = 𝑒 +

‖𝒖‖2

+‖𝒗‖2

2 (43)

so that

𝜕

𝜕𝑡𝜌𝐸∗ + 𝛁 ∙ [𝜌𝒗𝐸∗ + 𝑝𝒗 − �̆� ∙ 𝒗 − 𝜅𝛁𝑇] = 𝑞𝐻 + 𝜌𝒗 ∙ �̆� − 𝜌(𝒗 + 𝒖) ∙ �̇� (44)

or

𝜕

𝜕𝑡∫ 𝜌𝐸∗𝑑𝑉

𝑉

+ ∮ (𝜌𝐸∗𝒗 + 𝑝𝒗 − �̆� ∙ 𝒗 − 𝜅𝛁𝑇) ∙ 𝑑𝑺𝜕𝑉

= ∫ (𝑞𝐻 + 𝜌𝒗 ∙ �̆� − 𝜌(𝒗 + 𝒖) ∙ �̇�)𝑑𝑉.𝑉

(45)

This is an equation for energy conservation written in terms of generalized rothalpy and relative frame

velocities. Written instead in terms of velocities in Σ′, the rotation or spin �̆�, and the dynamics of 𝑂′

with respect to 𝑂 denoted by �̇̆̇� and �̇�, the equation is

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 11

𝜕

𝜕𝑡𝜌𝐸∗ + 𝛁 ∙ [𝜌𝒗𝐸∗ + 𝑝𝒗 − �̆� ∙ 𝒗 − 𝜅𝛁𝑇]

= 𝑞𝐻 + 𝜌𝒗 ∙ �̆� − 𝜌(𝒗 + 𝒖) ∙ �̇̆̇� − 𝜌(𝒗 + 𝒖) ∙ (�̇̆� × 𝒙)

= 𝑞𝐻 + 𝜌𝒗 ∙ �̆� − 𝜌𝒗 ∙ �̇̆̇� − 𝜌�̆̇� ∙ �̇̆̇� − 𝜌(�̆� × 𝒙) ∙ �̇̆̇� − 𝜌𝒗 ∙ (�̇̆� × 𝒙)

− 𝜌�̆̇� ∙ (�̇̆� × 𝒙) − 𝜌(�̆� × 𝒙) ∙ (�̇̆� × 𝒙)

(46)

The source terms may be interpreted as, from left to right from the second term, as the work associated

with: the gravity body force as seen in Σ′; the acceleration 𝑂′with respect to 𝑂, of the fluid in the

element, with velocity 𝒗 seen in Σ′; the acceleration of 𝑂′with respect to 𝑂 at the velocity of 𝑂′with

respect to 𝑂; the acceleration of 𝑂′with respect to 𝑂 at the spin velocity seen in Σ′; the Euler

acceleration due to spin-up or spin-down on the fluid in the element with velocity 𝒗; the velocity of

𝑂′with respect to 𝑂 and the Euler acceleration; and the Euler acceleration on the spin velocity seen in

Σ′. The Coriolis effect has correctly disappeared from equation (46).

3.7 Constitutive relations

In the equations of state of the gas, 𝜌, 𝑝 and 𝑇 transform as invariants. The constitutive relation for the

viscous stress tensor can be expressed in Σ as follows:

𝝉 = 𝜆(𝛁 ∙ 𝒗)𝑰 + 2𝜇[(𝛁𝒗) + (𝛁𝒗)𝑇], (47)

where 𝜆 is the bulk viscosity and 𝜇 is the dynamic viscosity. Both 𝜆 and 𝜇 may be treated as functions

of 𝜌, 𝑝 and 𝑇, which are invariant. As a second order tensor, 𝜏 is transformed by equation (6). The first

term becomes

�̆�1 = 𝑼𝑡 ∙ 𝜆(𝛁 ∙ 𝒗)𝑰 ∙ 𝑼. (48)

Using the scalar product definition in section 3.5, the inverse mapping 𝜑∗ in section 3.3, and the

definition of 𝛁 ∙ 𝚯 in section 3.5, it can be seen that 𝜑∗ (𝛁 ∙ 𝒗) = 𝛁 ∙ [𝑼𝑡 ∙ (𝜑∗𝒗)], and

�̆�1 = �̆�(𝛁 ∙ �̆�)𝑰 (49)

where �̆� = 𝜙∗𝜆 (section 3.3) and the spatial derivative is in Σ′. With the definition of interframe

velocity in equation (12), 𝛁 ∙ 𝒖 = 0, and

�̆�1 = �̆�(𝛁 ∙ 𝒗)𝑰. (50)

Similarly, the grad operator transforms as

𝜑∗ (𝛁𝒙𝒗) = 𝛁�̆�(𝜑∗𝒗). The second term becomes

�̆�2 = 2�̆� 𝑼𝑡 ∙ [(𝛁𝒗) + (𝛁𝒗)𝑇] ∙ 𝑼, (51)

and with the spatial derivatives written in Σ′

�̆�2 = 2�̆�[(𝛁�̆�) + (𝛁�̆�)𝑇]. (52)

For 𝒖 defined by equations (11) and (12) as the sum of a translation and a rotation, it is easily shown

that 𝛁 ∙ 𝒖 = 0 and that 𝛁𝒖 is antisymmetric. In writing 𝛁�̆� in terms of 𝛁𝒗, therefore, interframe

velocity gradients vanish, and

�̆� = �̆�(𝛁 ∙ 𝒗)𝑰 + 2�̆� [(𝛁𝒗) + (𝛁𝒗)𝑇

]. (53)

For problems in which relaxation times for local flow are important, it is useful to refer to an analysis

by de Groot and Mazur [62] in which pure rotation of a fluid is considered. In the inertial frame,

angular momentum is decomposed into a spin component (local rotation of fluid elements, as described

by 𝑼 in the present work) and a component describing bulk revolution about a centre of rotation (as

described by 𝒓 in the present work, in which completely arbitrary motion is of interest). In the pure

rotation case, de Groot and Mazur defined a rotational viscosity which can be related to a relaxation

time purely for local fluid element spin.

3.8 Conservation summary

The analysis in the sections above has been performed for several purposes. A point that is illustrated is

that for simulation of compressible flow in the relative frame, each source term added in coding must

be separately verified and validated in a numerical model constructed for the purpose.

Equations (21), (26), and (45) form a statement of the conservation laws in integral form, and

equations (22), (27) and (46) are the equivalent statements in differential form. Equation (53) gives the

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 12

transformed viscous stress tensor. In specific cases, dimensionless numbers can be devised to assist in

estimating the dominance of effects observed in the relative frame on a case-by-case basis.

4 Dimensionless numbers

The effects described above have been considered negligible in most cases of aeronautical interest. It

may however be necessary to determine the magnitude of these effects for missiles undergoing

“significant” acceleration. Several approaches can be followed to determine what accelerations are

“significant”. Of these, the simplest, and most approximate, is to determine the relative size of terms in

the equations of motion, and to understand regimes in which various effects are expected. This has

been done to a large extent for fluid phenomena in the circulation of the earth’s atmosphere, leading to

the formulation of the Rossby number and the investigation of Taylor-Proudman vortices in

incompressible flow within closed boundaries [5,6].

The approach of devising dimensionless numbers is useful in determining flow regimes, but is

essentially a linear approximation. Insofar as the approach can be used in any general way, rather than

in specific cases, the method is shown for the momentum equation here, and can be adapted to the

energy equation as well.

Following Boubnov and Golitsyn [8], a change of variables is made with respect to typical

scales as follows:

𝜑 = 𝜑0𝜑∗, 𝒙 = 𝐿𝒙∗, 𝛁 = L−1𝛁∗,

�̇� = �̇�0�̇�∗, 𝒗 = 𝑣0𝒗∗,𝜕2𝒓

𝜕𝑡2= 𝑎0

𝜕2𝒓∗

𝜕𝑡2 𝑎𝑛𝑑 𝝉 = 𝜈𝝉∗

(54)

where 𝜑 = 𝑡, 𝜔, 𝜌, 𝑝, 𝒈 or 𝒗, and where 𝜈 is a typical kinematic viscosity for the case, �̇�0 is a typical

angular acceleration of spin about 𝑂′, and 𝑎0 describes linear acceleration of 𝑂′ with respect to 𝑂, or

revolution, or both. An example of revolution without rotation is that of an aerofoil with constant pitch

attitude angle executing a circle about 𝑂, in which case the angle of attack varies during the revolution.

The scale factors are determined by the problem at hand. Rewriting the momentum equation

(27) as the following, omitting the asterisks, and multiplying by 𝐿

𝜌0𝑣02 in order to compare terms with

convection gives

𝐿

𝑣0𝑡0

𝜕

𝜕𝑡(𝜌𝒗) + 𝛁 ∙ 𝜌𝒗⨂𝒗 +

𝑝0

𝜌0𝑣02

∇ ∙ 𝑝𝑰 −𝜈

𝜌0𝑣02

𝛁 ∙ �̆�

= −𝐿2𝜔0

2

𝑣02

𝜌�̆� × ( �̆� × 𝒙) − 2𝐿2𝜔0

𝑣02

𝜌�̆� × 𝒗 −𝐿2�̇�0

𝑣02

𝜌�̇̆� × 𝒙

−𝐿𝑎

𝑣02

𝜌�̇̆̇� +𝐿𝑔0

𝑣02

𝜌�̆�

(55)

Using the definitions of the Strouhal number, 𝑆𝑡 =𝑣0𝑡0

𝐿 ; the Euler number, 𝐸𝑢 =

𝜌0𝑣02

𝑝0 ; the Reynolds

number, 𝑅𝑒 =𝐿𝑣0

𝜈 ; the Rossby or Kibel number, 𝑅𝑜 =

𝑣0

2𝜔0𝐿 ; and the Froude number, 𝐹𝑟 =

𝑣0

√𝐿𝑔, we

can write

𝑆𝑡−1

𝜕

𝜕𝑡(𝜌𝒗) + 𝛁 ∙ 𝜌𝒗⨂𝒗 + 𝐸𝑢−1∇ ∙ 𝑝𝑰 −

𝐿

𝑣0

𝑅𝑒−1𝛁 ∙ �̆�

= −𝐿2𝜔0

2

𝑣02 𝜌�̆� × ( �̆� × 𝒙) − 𝑅𝑜−1𝜌�̆� × 𝒗 −

𝐿2�̇�0

𝑣02 𝜌�̇̆� × 𝒙 −

𝐿𝑎0

𝑣02 𝜌�̇̆̇� + 𝐹𝑟−2𝜌�̆�

(56)

The first term and the Strouhal number characterize typical temporal behavior. The Euler term

compares pressure and inertial effects. The Reynolds number arises from the comparison of viscous

effect in translation (the normalization being different from the usual non-dimensionalisation leading to

the Reynolds number), while in rotational viscous problems the Ekman number, 𝐸𝑘 =𝜈

𝐿2𝜔0= 2𝑅𝑜 𝑅𝑒⁄ ,

arising from comparisons of the dominance of rotational and viscous effects, is used for example in

understanding the development of viscous boundary layers on the walls of rotating vessels [9]; if

rotation is near-rigid, then 𝐿𝜔0 ≈ 𝑣0 and the scale factors lead back to the Reynolds number.

The first source term is related to centrifugal effects viewed from the rotating frame Σ′, and

where a typical radius of rotation R applies for rigid body rotation the coefficient is of the order 𝐿2 𝑅2⁄

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 13

and the effect is easily visualised. The second term is the familiar Coriolis term characterised by the

Rossby number. The third term is the spin-up or spin-down term. In the formulation provided here, 𝜔0

and �̇�0 are linearly related by the typical time scale, but the time scale of angular acceleration can be

decoupled from 𝑡0 if this is appropriate for the problem. The final terms establish the equivalence of

translational acceleration 𝑎 = �̈� and gravity 𝑔 in the relative motion of Σ′ and Σ. We term the factor

𝑄 =𝐿𝑎0

𝑣02 the acceleration (or deceleration) parameter

1 and note that it is the counterpart of 𝐹𝑟−2. It

should be noted that 𝒓 may describe revolution at a constant attitude angle, while 𝝎 is spin and governs

attitude changes. A set of parameters can now also be derived for the Euler acceleration and cross-

coupling terms in equation (46) for specific circumstances.

Other dimensionless parameters for fluids still emerge from the Navier-Stokes equations and

must be taken into account, most importantly including the Mach number M.

5 Illustrations

5.1 Linear acceleration

The works of Roohani and Skews [41,43,44] focus on acceleration and deceleration effects when the

acceleration is parallel to the velocity of the object. In this case, the conservation equations in the

relative frame Σ′ are as follows, given that 𝝎 = 0, and therefore 𝑼 = 𝑰. Conservation of mass remains

𝜕𝜌

𝜕𝑡+ 𝛁 ∙ (𝜌𝒗) = 0. (57)

Conservation of momentum becomes, from (26) and (27)

𝜕

𝜕𝑡(𝜌𝒗) + 𝛁 ∙ [𝜌𝒗⨂𝒗 + 𝑝𝑰 − �̆�] = −𝜌�̇̆̇� + 𝜌�̆� (58)

The equation for energy conservation written in terms of rothalpy and relative frame velocities in Σ′ is

now (45), (46)

𝜕

𝜕𝑡𝜌𝐸∗ + 𝛁 ∙ [𝜌𝒗𝐸∗ + 𝑝𝒗 − �̆� ∙ 𝒗 − 𝜅𝛁𝑇] = 𝑞𝐻 + 𝜌𝒗 ∙ �̆� − 𝜌𝒗 ∙ �̇̆̇� + 2𝜌�̆̇� ∙ �̇̆̇�. (59)

With 𝝎 = 0, �̆� = 𝒈 by equations (2) and (5), and �̆̇� = �̇� , �̇̆̇� = �̈� by (9) and (1). The acceleration

parameter can be written as 𝑄 =𝐿�̈�

�̇�2 .

Roohani and Skews studied the effects of acceleration and deceleration on biconvex, diamond,

NACA0012, NACA2412, and RAE2812 aerofoils. Parameters for their cases are shown in Table 1. For

comparison, maximum thrust during the launch of a typical 90 kg Vth generation SRAAM (short range

air-to-air missile) is or the order of 20 000 N. For present purposes we assume the altitude to be of the

order of 30 000 feet and the Mach number in the region of 1.5.

Parameter Aerofoils,

subsonic

range

Aerofoils,

transonic

range

SRAAM,

supersonic

range

Typical length 𝐿 1 𝑚 1 𝑚 3 𝑚

Typical velocity 𝑣0 100 𝑚𝑠−1 300 𝑚𝑠−1 464 𝑚𝑠−1

Mach number M 0.3 1 1.5

Typical acceleration 𝑎0 1041𝑚𝑠−2 1041 𝑚𝑠−2 980 𝑚𝑠−2

Acceleration parameter 𝑄 =𝐿𝑎0

𝑣02 0.1 0.01 0.01

Freymuth start-up time √𝐿/𝑎 0.03 𝑠 0.03 𝑠 0.06𝑠

Table 1 Typical parameters for aerofoils and air-to-air missiles. Data for aerofoils is from

Roohani and Skews [44].

1 For linear acceleration this is consistent with cosmological usage.

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 14

Significant effects on aerofoil lift and drag were observed [44] in the subsonic range, and we

deduce that 𝑄~0.1 may be a reasonable indicator in this range. In the high velocity ranges 𝑄 is lower,

but shock wave effects dominate flow history in this range, as described under section 2, the Mach

number M ~ 1. For supersonic missiles, studies using computational fluid dynamics remain to be done.

The Freymuth start-up time is independent of velocity and should be shorter under more significant

acceleration; together with the Mach number it provides a consistent predictor for the aerofoils.

The effects on drag are significantly different for acceleration, �̈� �̇�⁄ > 0, and deceleration

�̈� �̇�⁄ < 0. This obviously important parameter determines whether wake effects and shocks propagate

faster than the aerofoil itself. In this case, �̈� �̇�⁄ < 0, shocks may overtake an aerofoil [44], and stand-off

bow shocks may exist at subsonic body velocities [44,52]. For transonic acceleration or �̈� �̇�⁄ > 0,

shocks on convex surfaces are likely to be found forward of the steady state position [44]. In terms of

flow history as defined above, this is due to persistent features of the flow field at an earlier time and a

lower Mach number. Similarly, for deceleration or �̈� �̇�⁄ < 0, transonic shocks will be found aft of the

steady state position on an airfoil surface and may be attached to the trailing edge at lower velocities

than expected on steady-state grounds.

5.2 Flows confined by axisymmetric boundaries

The smooth, controlled combustion of fuel in solid rocket motors is a subject of considerable practical

importance in trajectory predictability and control. A set of parameters describing the flow regimes

may prove useful. A start may be made by considering parameters governing the development of

Taylor-Proudman vortices in incompressible fluids in closed rotating containers [9]. While rocket

motors are open containers, solid rocket charges are internally shaped, mass is variable, the fluids are

compressible, and temperatures are such that real gas effects must be considered in flow, some insight

may still be gained into the dominance of rotating effects by progressively working from simple

assumptions such as rotation that may be almost rigid.

Taylor-Proudman vortices are possible when incompressibility and boundary conditions

together make internal flow independent of the coordinate parallel to the rotational axis, which we will

take to be 𝑧, and a perturbation along the axis will cause perturbation of streamlines independent of z

[5,6] and will occur when rotational effects dominate, for example when 𝑅𝑜 ~ 12. Similar dynamics

govern the establishment of rigid rotational flow near a spinning disk, for example a chamber end-wall,

when the angular velocity is perturbed, or when the wall is moved along the rotational axis [9]. Viscous

effects during spin-up or spin-down include the development of an Ekman layer in a plane normal to 𝑧

and the propagation of a front between rotating and static flow towards, or away from, the axis [9] on

time scales of the order of 𝜔0−1, and the establishment of rigid body flow on the typical time scale

𝜔0−1 𝐸𝑘−1

2⁄ (taking 𝜈 as that for dry air for approximate purposes).

Some data for rockets may be found in the literature from which the typical values have been

deduced. Apache and Cajun sounding rockets were described by Uselton and Carman [63]; the radius

was 59 mm and rotation rates were 1.5 to 32 rad s-1

. Marquardt et al. [64] provided data on highly

manoeuvrable missiles, based on radius 59 mm and rotation rates of order 100 rad s-1

, but for which

only limited information is available. Unguided fin-stabilised artillery rockets with 122 mm body

diameter were covered by Khalil et al. [65] with typical muzzle spin rates of 100 rad s-1

.

Low spin Medium spin High spin

Spin rate 𝜔0 1.5 𝑟𝑎𝑑 𝑠−1 32 𝑟𝑎𝑑 𝑠−1 100 𝑟𝑎𝑑 𝑠−1

Radius r 59 𝑚𝑚 59 𝑚𝑚 61 𝑚𝑚

Ro, rigid .5 .5 .5

Ek 3.1x10-3

1.4x10-4

4.3x10-5

Ekman layer time .67 𝑠 .03 𝑠 .01 𝑠

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 15

𝜔0−1

Spin up time

𝜔0−1 𝐸𝑘−1

2⁄

12.0 𝑠 2.6 𝑠 1.5 𝑠

Table 2 Internal flow parameters for the assumption of almost-rigid flow. Times are given by

Greenspan [9]; parameters for low and medium spin conditions are sourced from Uselton and Carman

[63] and for high spin conditions were given by Khalil et al. [65].

In the low spin and medium spin cases, if the chamber were to be considered to be closed and

incompressible inviscid flow were to be assumed, Taylor-Proudman effects might be expected. In the

high spin case, Taylor-Proudman effects might be expected and the Ekman layer would be established

before burnout since the burn time is 1.8 s. If Taylor columns were to be established, the implication is

that deformities in the grain would influence the flow field parallel to the spin axis along the length of

the internal space. These are very considerable approximations. However, they illustrate the fact that

some remarks may be made about rotational effects which may affect grain burn under certain

circumstances.

While the examples above illustrate well-known parameters, the aim in introducing them here is

to add to the basis on which useful parameters for arbitrary manoeuvre can be built.

6 CFD implementation

A few relevant studies have already been undertaken with Computational Fluid Dynamics models and

in the following paragraphs the methodologies of different implementations are discussed. ANSYS

FLUENT and STAR-CCM+ are two commercial CFD software packages, allowing the use of multiple

solvers, compressibility models, moving mesh capabilities and a large range of turbulence models.

They are both useful in investigations into the unsteady aerodynamics of accelerating bodies. However,

the approach that is used with each code is completely different.

STAR-CCM+ is of interest because of its use of overset meshing, also known as Chimera

meshing [66,67]. This is a capability where a mesh overlaps, or is superimposed on, another mesh in

the computational domain. The two meshes are coupled and a reference system is incorporated to

interpolate the values of the cell variables of each mesh through the use of algebraic equations. This

meshing option is most commonly implemented in cases with moving bodies. The background mesh

representing the fluid is held stationary and the overset mesh moves with the moving body, thus

providing an inertial reference frame. The velocity of the overset mesh can either be characterized by a

user defined function or by a set of tabulated values. One drawback of this approach is that for large

unidirectional velocities a very large background domain is required resulting in excessive

computational time. However, it is well suited to problems analysing combined rotational and

translational acceleration over short distances.

ANSYS FLUENT [46] uses a non-inertial reference frame where a single mesh system is used.

In order to model an accelerating body the object is held stationary and the fluid in every cell in the

meshed domain is accelerated simultaneously with equal acceleration across the domain. User defined

functions are required in order to define the acceleration at the boundary of the domain and to add

source terms to the momentum and energy equations to ensure that the same acceleration is

implemented within the meshed domain. This approach is useful for analysing problems with large

unidirectional velocities and accelerations as the size of the meshed domain remains fixed resulting in

reduced computational time compared to inertial reference frame systems.

A different approach to implementation has been taken within the codes EURANUS [68] and its

successor Edge [31]. Edge is an unstructured dual-grid compressible flow solver on arbitrary elements

based on a node-centred finite volume technique. The code was designed for aeroelastic application

and is based on moving grids, and this greatly simplifies the adaption of the code to providing a rigid

moving grid directly implemented in the absolute frame Σ. No source terms are needed, or need to be

verified, in this implementation. Simulations in this frame have been tested and validated [1]. The

major advantage of rigid-grid absolute frame CFD is that the domain is the size of a traditional CFD

grid for a missile, extending only to far field boundaries of the order of 20 typical lengths away from

the object. It was necessary to provide the basic transforms, equations (7) and (9), for use with results,

since most velocity flow fields in the absolute frame appear unfamiliar.

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 16

Arbitrary Lagrangian-Eulerian (ALE) methods are based on a third frame of reference which

may, or may not, be coincident with the Lagrangian or Eulerian frame [69], and have proved useful in

finite-element models of deforming materials, free surfaces, and fluid-structure interactions [70]. While

they have been applied to prediction of dynamic derivatives [71], instances of the use of ALE models

for long trajectory manoeuvre or acceleration have not been encountered.

7 Discussion

The purposes and results of writing the conservation equations in the forms above are as follows.

The general transform is needed for the interpretation of CFD results. Conventionally, CFD has

been performed in the relative frame, and results are inspected in this frame in a familiar form. If

computation is performed in the absolute frame using a moving grid, and then displayed on that grid,

𝒗 = 0 at the boundaries in stagnant air in 𝛴 while 𝒗 ≠ 0 at stagnation points (defined as points at

which the flow velocity equals the wall velocity). Potential confusion can be overcome by inspecting

flow fields in both frames.

This set of equations allows the identification of flow dynamics and features in both frames. For

example, Gledhill et al. [1] shows that missile strake vortex cores deflect from their steady-state paths

towards the turn centre in a constant-pitch turn and may disrupt expected fin dynamics. In the relative

frame this can be interpreted as a centrifugal effect, displacing higher-density fluid away from the turn

centre. Similarly, awareness of the well-known dynamics of rotating flow may assist in building an

understanding of flow under accelerated conditions as illustrated by the spinning rocket examples

above.

The Navier-Stokes equations have been displayed in a form that lends itself to finite volume

implementation in CFD, in both absolute and relative frames. The source terms are explicitly listed and

interpreted for the relative frame. The effects of each source term implemented need to be verified and

validated.

A full set of dimensionless numbers has been collected for the momentum equation applied to

arbitrary manoeuvre in order to guide estimation of regimes in which significant acceleration effects

may appear. The Mach number is an important addition to this set. Examples of significant effects have

been illustrated through published results for linearly accelerating and decelerating aerofoils, and it is

predicted that air-to-air missiles are within the acceleration range at launch at which effects are

significant.

In unidirectional linear acceleration, the acceleration parameter 𝑄 = 𝐿𝑎0 𝑣02⁄ is derived directly

from the momentum equation in the relative frame. This parameter does not take compressible effects

into account but does indicate the size of linear acceleration terms in comparison with convection. The

obvious importance of acceleration or deceleration, �̈� �̇�⁄ , has been illustrated with reference to the

models of Roohani and Skews [45] and the experiments and simulations of Saito et al. [52].

A few examples of dimensionless parameters in very simplified configurations have been given.

For angular acceleration or for constant angular velocity, these equations reduce to those in the

literature of rotating flow, and the case of the Taylor-Proudman theorem follows.

Turbulence models have been developed over the years based on observations in constant

velocity wind tunnel cases, and can therefore be regarded as holding in the inertial frame. Using the

formulation developed here, the equations for turbulence models can easily be transformed into a non-

inertial frame, but the question of how turbulence behaves for rapidly accelerating objects has yet to be

investigated and answered.

8 Conclusions

This paper has presented the formulation of continuity, momentum, and energy equations for

fluids in a way that enables manipulation of these equations across absolute inertial and body-fixed

relative frames for arbitrary acceleration.

These expressions are now in a form that allows selected terms to be transformed clearly

between non-inertial and inertial frames, and explicitly shows the coordinate system in which each

vector or tensor is expressed. The transforms allow CFD results to be shown in the relative frame,

which is the frame traditionally used in CFD and therefore often used for flow field interpretation, and

the absolute frame, in which computation is simpler but results are less intuitively easy to construe.

Both gravity and linear acceleration are formally included.

Gledhill, Roohani, Forsberg, Eliasson, Nordström and Skews 17

The derivation of a number of dimensional constants has been demonstrated in order to provide

some guidance on regimes in which accelerating flow would be expected to be significant.

Acknowledgements

This document is partially a result of a research effort funded by the DRDB in terms of Armscor

Orders KT466921, KT528944 and KT470887 before 2006. I. Gledhill wishes to acknowledge with

gratitude the joint project set up by the CSIR that made this work possible, and the great hospitality of

the University of the Witwatersrand School of Mechanical, Industrial and Aeronautical Engineering

without which it could not have been completed. H. Roohani acknowledges the contribution of the

Fluent and CD-adapco teams in accomplishing the simulations.

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